## Abstract

The surface plasmon (SP) coupling behaviors of an embedded light emitter or radiating dipole in GaN with a surface Ag nanoparticle (NP) in four structures of different added dielectric geometries, including an extended dielectric interlayer (DI) and a DI of a finite width between the Ag NP and GaN, a dielectric coating on the Ag NP, and no dielectric addition, are numerically compared. Either an added DI or dielectric coating can lead to the blue shift of localized surface plasmon (LSP) dipole resonance peak or the spectral peak of radiated power enhancement ratio with respect to that of the structure without dielectric addition. A smaller dielectric refractive-index or a larger dielectric thickness results in a larger blue-shift range. Under the condition of the same dielectric refractive-index and thickness, the structure of a DI with a finite width leads to the largest blue-shift range, followed by the structure of an extended DI and then the structure of a dielectric coating. In a practical application, for a given emission wavelength of a blue-emitting quantum well, the emission enhancement effect through SP coupling depends on the LSP resonance strength at this wavelength. Our study also shows that although the LSP resonance peak can be blue-shifted by reducing the size of a surface Ag NP, its SP coupling strength is dramatically reduced. Adding a DI or dielectric coating is a more practical approach for shifting the major LSP resonance mode of a surface Ag NP from the green into blue range.

© 2015 Optical Society of America

## 1. Introduction

Besides the enhancement of internal quantum efficiency (IQE) [1–5], the important advantages of surface plasmon (SP) coupled light-emitting diode (LED) include the reduction of its efficiency droop effect [6–8]. The enhancement of IQE is particularly effective in an LED of low intrinsic IQE [9], such as those in the green-yellow and ultraviolet spectral ranges. The efficiency droop effect is still a problem not completely solved yet in the whole LED spectral range, including blue LED, which has very high IQE already. Therefore, SP coupling in a blue LED is useful for reducing its efficiency droop effect. A few attributions have been proposed for the efficiency droop effect in an LED, including the widely discussed Auger recombination [10–12] and current overflow [13–15]. In either attribution of Auger recombination and current overflow, the fundamental cause is the high carrier density in a quantum well (QW) when injection current level is high. With the SP coupling process, because of the enhanced emission rate of the SP-QW coupling system, the carrier density in a QW can be effectively decreased, resulting in the reduction of the efficiency droop effect.

Among various metals for inducing the SP-coupling process, Ag is the one most suitable for the application in the blue-green spectral range. Also, among various Ag nanostructures for inducing SP coupling in an LED, surface Ag nanoparticle (NP) represents a choice of simple fabrication and low cost. On GaN, which has a refractive-index around 2.4 in the visible range, the dipole and higher-order resonance modes of localized surface plasmon (LSP) of an Ag NP can cover the spectral range from violet through yellow. Generally speaking, the LSP dipole resonance mode has larger resonance strength and a lower dissipation rate, when compared with a higher-order resonance mode, leading to a stronger emission enhancement effect. However, the LSP dipole resonance mode of an Ag NP on GaN is usually spectrally located in the green-yellow range such that the use of surface Ag NP for inducing SP coupling in the blue range is ineffective [16, 17]. Experimental efforts have been made to blue-shift the LSP dipole resonance mode into the blue range by reducing the Ag NP size [18]. However, so far no LED was really fabricated with this technique. Recently, this research group has experimentally demonstrated a new technique for effectively blue-shifting the LSP dipole resonance mode and hence causing a significant SP coupling effect in a blue LED by inserting a thin dielectric interlayer (DI), which has a refractive-index lower than that of GaN, between Ag NPs and p-GaN [19]. With either a lower refractive-index or a larger thickness in the DI, the blue-shift range of the LSP resonance peak is larger. By properly choosing the refractive-index and thickness of the DI, the LSP resonance strength at the QW emission wavelength can be maximized for optimizing the SP coupling effect.

Although the methods for blue-shifting the LSP resonance peak of a surface Ag NP have been demonstrated, the detailed SP coupling behaviors with different DI geometries (e.g., finite or infinite DI width) have not been studied yet. In particular, in terms of the blue-shift range and LSP resonance strength for SP coupling, whether the method of reducing Ag NP size or the technique of inserting a DI is more effective is an important question to answer for practical application. Also, it was demonstrated that Ag NPs coated with SiO_{2} shells could be useful for inducing SP coupling when they are placed on an LED [20–22]. Based on the SP coupling theory, the Ag/SiO_{2} core-shell NP is similar to the aforementioned structure of a surface Ag NP with a DI. The difference of SP coupling effect between such a core-shell Ag NP and an Ag NP with a DI deserves study for optimizing the performance of an SP-coupled LED. In this paper, we numerically compare the SP coupling behaviors of a surface Ag NP with a radiating dipole embedded in a GaN layer under the conditions of various dielectric structures and different Ag NP sizes. The dielectric structures include an extended DI, a finite-width DI under the Ag NP, and a dielectric coating on the Ag NP. Different dielectric refractive-indices and thicknesses are considered. In section 2 of this paper, the problem geometries and simulation method are described. The simulation results under various conditions are shown in section 3. Then, discussions about the numerical results are made in section 4. Finally, conclusions are drawn in section 5.

## 2. Problem geometries and simulation method

Figures 1(a)-1(d) show the four problem geometries under study in this paper, which are referred to as the structures of samples A-D, respectively. In each sample structure, an embedded radiating dipole, which is represented by a thick horizontal arrow, at a fixed depth, *d*, in a GaN layer couples with the LSP modes induced on a surface Ag NP in air. For simulating a real Ag NP on a flat surface [23], the Ag NP is assumed to be a truncated ellipsoid with the lengths of its semi-axes at *a* and *b*, respectively, and the truncated range equal to *b* - *t*. In Fig. 1(a) for sample A, the Ag NP directly contacts the GaN surface. In Fig. 1(b) for sample B, between the Ag NP and GaN, a thin film of DI (infinitely large width) is inserted with the thickness at *h*. Then, in Fig. 1(c) for sample C, the lateral range of the DI is limited to be only in the area below the Ag NP. The structure of sample B shown in Fig. 1(b) can be used for SP-coupled LED implementation when the DI is conductive in direct-current operation, such as GaZnO or InSnO [19]. That of sample C shown in Fig. 1(c) needs to be used for SP-coupled LED implementation when the DI is an insulator, such as SiO_{2}. In this situation, the area of the DI must be limited; otherwise, the device resistance can be tremendously increased. The DI of a limited area can be fabricated with a dry-etching process. The structure of sample D shown in Fig. 1(d) represents an Ag/SiO_{2} core-shell NP on GaN. Although such a core-shell NP is not necessarily an ellipsoid in shape, for easily comparing with the results of the other three samples, the structure with a conformal dielectric shell of uniform thickness on a truncated Ag ellipsoid, as shown in Fig. 1(d), is adopted for numerical study. It is noted that the origin of the used rectangular coordinate system is set at the center of the Ag-NP base in each sample.

The three-dimensional numerical simulations are carried out with the assistance of the commercial software COMSOL, which is based on the numerical algorithm of the finite-element method. For simulation, the computation domain is chosen to be a spherical region. It contains the radiating dipole embedded in the lower-hemisphere of GaN and the truncated Ag ellipsoid together with the dielectric structure in the upper-hemisphere of air. To simulate the infinite extension of the background environment, a spherical layer, known as the perfectly matched layer, is placed right outside the computation domain. We first compute the radiated electromagnetic field (the incident field) of a radiating dipole situated in a homogeneous (GaN) spherical background region. Then, the total field is calculated for the real problem geometry including the radiating dipole, Ag NP, and dielectric structure. By subtracting the incident field from the total field, we can obtain the scattered field, which is to be used for computing the feedback effect from the LSP resonance on the dipole radiation behavior. With the available scattered field, the optical Bloch equations are solved to find the resultant strength and orientation of the modified dipole [5]. Based on this modified dipole, the final total electromagnetic field as well as the total radiated power can thus be calculated numerically. For numerical computations, three Ag NP sizes are considered, including the large (L) NP with *a* = 30 nm, *b* = 40 nm, and *t* = 20 nm, the medium (M) NP with *a* = 22.5 nm, *b* = 30 nm, and *t* = 15 nm, and the small (S) NP with *a* = 15 nm, *b* = 20 nm, and *t* = 10 nm. In other words, the ellipsoidal aspect ratios of the three NPs are the same. Also, the dielectric constant of GaN is fixed at 5.76, and experimental data are used for the dielectric constant of Ag [24]. Throughout this paper, the depth of the embedded radiating dipole in GaN, *d*, is fixed at 50 nm. In presenting the simulation data, the normalized absorbed and radiated powers are plotted as functions of wavelength. They are normalized to the total radiated power-in the case of no Ag NP and no dielectric structure, i.e., the structure of sample A [see Fig. 1(a)] without the Ag NP. The spectral peaks of the normalized absorbed power indicate the spectral positions of LSP resonance modes. The spectra of normalized radiated power show the emission enhancement levels at different wavelengths.

## 3. Simulation results

Figure 2 shows the spectra of normalized absorbed power of the three Ag NP sizes in the structure of sample A (indicated by the capital letters within the parentheses of the labels in the figure) when the radiating dipole is laterally located at *x* = 0. Here, the major and minor peaks in each curve correspond to the dipole and quadrupole LSP resonance modes, respectively [17]. One can see that as Ag NP size is reduced, the spectral peak of the dipole resonance mode blue-shifts. However, the shift range is quite small. Similarly, the spectral peak of the quadrupole resonance mode does not shift significantly. Also, the normalized absorbed power level is significantly reduced when the Ag NP dimension is reduced by 50%, indicating that the LSP resonance strength decreases drastically with reducing Ag NP size. Figure 3 shows the normalized radiated power spectra corresponding to the cases in Fig. 2. Again, the major and minor peaks in each curve correspond to the dipole and quadrupole LSP resonance modes, respectively. However, the spectral peak wavelength of the individual resonance mode is slightly shorter than that of the absorption spectrum. Here, the peak level of normalized radiated power is reduced from 2.69 to 1.25 when the Ag NP dimension is reduced by 50%, confirming the drastic reduction of SP coupling effect in decreasing metal size. From the results in Figs. 2 and 3, one can conclude that to blue-shift the LSP resonance mode for effective SP coupling in the blue range, the reduction of metal size is not an effective approach. In reducing NP size, although a larger planar NP density can somewhat compensate the reduction of the SP coupling strength of an individual NP, the small blue-shift range of SP-coupling spectrum still makes this technique unattractive.

Figure 4 shows the spectra of normalized absorbed power when Ag NP S (the small NP) is used in the structure of sample B under the four conditions of different thicknesses (*h* = 5 and 10 nm, indicated by the number before the slash in a curve label) and different refractive-indices (1.8 and 1.5, indicated by the number after the slash in a curve label). For comparison, the corresponding result in the structure of sample A (No-DI) is also plotted in Fig. 4. Here, one can see that with the DI, both the dipole and quadrupole LSP resonance peaks significantly blue-shift into the blue range. The blue-shift range increases with increasing DI thickness and with decreasing DI refractive-index. Also, the LSP absorption level or resonance strength decreases with increasing DI thickness (decreasing DI refractive-index) when DI refractive-index (thickness) is fixed. Although the LSP resonance strength is reduced by adding the DI, it is still quite strong for effective SP coupling. Figure 5 shows the spectra of normalized radiated power corresponding to the cases in Fig. 4. Here, one can see that in the blue range, emission enhancement can be observed by adding the DI in the structure of sample B. However, the enhancement ratio is quite small. Figure 6 shows the spectra of normalized radiated power similar to those in Fig. 5, except that the large Ag NP (L) is used in the structure of sample B. Compared with the results of the small Ag NP in Fig. 5, the blue-shift ranges of the corresponding DI conditions with the large Ag NP are smaller. However, the emission enhancements in sample B are not much reduced from that of sample A, as shown in Fig. 6. In the case of 10/1.5(B), at 450 nm in wavelength, the emission enhancement ratio can still be as large as ~2.2, which is about 82% that in the case of No-DI(A) at 565 nm. From the results in Fig. 6, one can conclude that to effectively blue-shift the LSP resonance mode for strong SP coupling in the blue range, the addition of a DI is a useful approach.

If an insulator is used as the DI, the structure of sample C needs to be used for reducing device resistance. Figure 7 shows the spectra of normalized radiated power of the cases with DI refractive-index at 1.5 for the comparison between samples B and C. Here, one can see that when the DI coverage is limited to the base of the Ag NP, the SP coupling spectra are further blue-shifted with slightly weaker coupling strengths. In particular, as shown in Fig. 7, when the DI thickness is 5 nm in sample C, the enhancement peak of radiated power is shifted to ~450 nm with the peak level higher (by ~9.2%) than the corresponding value in sample B when the DI thickness is 10 nm. Therefore, a finite DI width can lead to a more favorable emission enhancement result by properly choosing the DI thickness when SP coupling at a given QW emission wavelength is needed.

The results shown above are obtained when the radiating dipole is located right below the center the Ag NP, i.e., at *x* = 0. In this situation, the SP coupling behavior is independent of the dipole orientation in the *x*-*y* plane due to the circular symmetry of the Ag NP. When the dipole position is laterally shifted, the SP coupling results with an *x*- and *y*-oriented radiation dipoles become different. Figures 8 and 9 show the spectra of normalized radiated power when the *x*- and *y*-oriented radiating dipoles, respectively, are located at different positions along the *x*-axis for samples B and C. Here, *x _{0}* is equal to one-quarter the diameter of the circular base of the Ag NP, as shown in Figs. 1(a)-1(d). It is given by

*x*=

_{0}*a*[1-(

*t*/

*b*)

^{2}]

^{1/2}/2. For the results in Figs. 8 and 9, we use the large NP as defined before, DI thickness

*h*at 10 nm, and DI refractive-index at 1.5. In either Fig. 8 or 9, one can see that the peak level of radiated power spectrum decreases with increasing

*x*in either sample B or C. However, the decreasing trend of sample C is weaker, when compared with sample B, such that the peak level difference of normalized radiated power between samples B and C decreases with increasing

*x*. This is true for either

*x*- or

*y*-oriented dipole. In Fig. 9 for a

*y*-oriented dipole, the spectral peak position keeps red-shifting with increasing

*x*. However, in Fig. 8 for an

*x*-oriented dipole, the red or blue shift of the spectral peak position in either sample B or C depends on the

*x*coordinate of the dipole.

In Figs. 10 and 11, we show the spectra of normalized radiated power when the *x*- and *y*-oriented radiating dipoles, respectively, are located at different positions along the *x*-axis for sample D. For comparison, the corresponding results of sample A are also shown in Figs. 10 and 11. Here, again, we use the large NP as defined before, DI thickness *h* at 10 nm, and DI refractive-index at 1.5. In either Fig. 10 or 11, we can first compare the curve of *x* = 0(D) with those of *x* = 0(B) and *x* = 0(C) in either Fig. 8 or 9. We can see that the blue-shift range of the spectral peak of normalized radiated power, with respect to that of sample A, in sample D is smaller than that of either sample B or C. Although the peak level of radiated power in sample D is higher than that in sample C, it is lower than that in sample B. When *x* is nonzero, the variation trends of radiated power peaks for *x*- and *y*-oriented dipoles in sample D are similar to their counterparts in sample C. The spectral peak keeps red-shifting with increasing *x* for a *y*-oriented dipole. As *x* increases, the spectral peak of an *x*-oriented dipole is blue-shifted first and then red-shifted.

## 4. Discussions

By comparing the spectral peak positions of curves *x* = 0(B) and *x* = 0(C) in Fig. 8 and that of curve *x* = 0(D) in Fig. 10, we can see that with the same Ag NP geometry and dielectric refractive-index and thickness, sample C leads to the largest blue-shift range in the spectral peak of radiated power, followed by sample B and then sample D. From the corresponding spectral peak of sample A at 565 nm, the spectral peaks of samples B, C, and D are blue-shifted to 453, 429, and 469 nm, respectively. To understand the cause for the variation of blue-shift range, we plot the distributions of normalized field magnitude (in log scale) of samples A-D in the *x*-*z* plane around the Ag NP and radiating dipole at 565, 453, 429, and 469 nm in Figs. 12(a)-12(d), respectively, which correspond to the individual spectral peaks of radiated power, when the Ag NP geometry and dielectric refractive-index and thickness are the same as those for Figs. 8-11. In these figures, thin lines are plotted for showing the boundaries between different materials. The radiating dipoles at *x* = 0 are represented by the horizontal (pink) arrows. Within a circle around the radiating dipole, field magnitude is not plotted because it is extremely large there. In Fig. 12(a) for sample A, we can see that the field distribution is concentrated around the base of the Ag NP, particularly around its edges. As a DI is added, the field distribution extends upward to the sidewall of the Ag NP. This upward-extension trend of field distribution is particularly clear in sample C, followed by samples B, then D, and then A. Such field distribution variations in changing the dielectric structure can also be seen in charge distributions. In Figs. 13(a)-13(d), we show the surface charge distributions of samples A-D corresponding to the field magnitude distributions in Figs. 12(a)-12(d), respectively. The color coding for Figs. 13(a)-13(d) is shown right to Fig. 13(d). The individual wavelengths for plotting those figures correspond to the LSP dipole resonance modes in samples A-D. In such a dipole resonance mode, electrons oscillate between the two one-halves of the Ag NP. Although the charges are mainly distributed around the base of the Ag NP, their distribution more or less extends upward to the sidewall of the NP. Among the four samples, again, the upward extension range is the largest in sample C, followed by samples B, then D, and then A. For a clearer comparison between samples B and C, in Figs. 13(e) and 13(f), we show the same charge distributions as those in Figs. 13(b) and 13(c), respectively, except that the color coding range is decreased, as shown in the color coding right to Fig. 13(f). From Figs. 13(e) and 13(f), we can see better the difference of charge density on the NP sidewall between samples B and C. The upward extension range of charge distribution in sample C is indeed the largest.

The upward extensions of LSP mode field and surface charge distributions imply that the resonance of this LSP mode is more influenced by the surrounding medium of the Ag NP sidewall. A larger upward-extension range leads to a stronger influence on the resonance behavior, particularly the resonance wavelength, by the surrounding refractive-index of the Ag NP (unity in air). Because a smaller surrounding refractive-index results in a shorter LSP resonance wavelength, a larger upward-extensions range of LSP mode field or surface charge distribution corresponds to the condition of a shorter LSP resonance wavelength. Since the upward-extension range is the largest in sample C, followed by samples B, then D, and then A, the LSP resonance wavelengths of the four samples must follow the same variation order. This variation order is indeed consistent with the results at *x* = 0 shown in Figs. 8-11. It is noted that for a given QW emission wavelength in an LED, the optimized condition for maximizing SP coupling effect depends on the LSP resonance strength or the radiated power enhancement ratio at the QW emission wavelength. A larger blue-shift range of LSP resonance wavelength in adding a dielectric structure does not necessarily lead to a larger radiated power enhancement ratio.

In Figs. 8 and 10, one can see that in sample A, C, or D, there is a clear gap in the radiated power spectra of an *x*-oriented dipole between the conditions of *x* = *x _{0}* and

*x*= 2

*x*. However, this behavior is unclear for a

_{0}*y*-oriented dipole, as shown in Figs. 9 and 11. As shown in Figs. 1(a)-1(d),

*x*= 2

*x*corresponds to the edge of the Ag NP base. For an

_{0}*x*-oriented dipole, when the dipole position is moved away from

*x*= 0 along the

*x*-axis, the field magnitude distribution of the excited LSP dipole resonance mode becomes asymmetric with respect to the

*y*-

*z*plane, i.e., asymmetric in a plane containing the electron oscillation axis. In this situation, on one side of the

*y*-

*z*plane, the surface charge or mode field is distributed on a large sidewall-surface area surrounded by air with a low density. Figures 14(a)-14(d) show the normalized field magnitude distributions in log scale (in the

*x*-

*z*plane) of samples A-D, respectively, around the Ag NP and an

*x*-oriented radiating dipole at 540, 458, 425, and 464 nm, which correspond to the individual spectral peaks of radiated power of the LSP dipole resonance mode, when

*x*= 2

*x*. Here, we can see the asymmetric field distribution in each sample structure. On the other hand, for a

_{0}*y*-oriented dipole, the field distribution of the excited LSP dipole resonance mode is always symmetric with respect to the

*x*-

*z*plane, i.e., symmetric in a plane containing the electron oscillation axis, when the dipole position is moved away from

*x*= 0 along the

*x*-axis. The similar field magnitude distributions in the

*y*-

*z*plane for a

*y*-oriented dipole at

*x*= 2

*x*in samples A-D at 570, 461, 430, and 470 nm are shown in Figs. 14(e)-14(h), respectively. Here, we can see the symmetric field distributions in all samples. It is speculated that such a symmetric LSP dipole resonance mode in Figs. 14(e)-14(h) can be stronger than an asymmetric mode shown in Figs. 14(a)-14(d). Therefore, as the lateral coordinate of a radiating dipole increases, the radiated power of a

_{0}*y*-oriented dipole decays more slowly than that of an

*x*-oriented dipole.

In Figs. 9 and 11, one can see that as the *x*-coordinate of a *y*-oriented dipole increases, the spectral peak of radiated power keeps red-shifting. However, the shift trend of an *x*-oriented dipole is more complicated. In Figs. 8 and 10, we can see that as the *x*-coordinate of an *x*-oriented dipole increases, the spectral peaks of radiated power are first blue- and then red-shifted in samples C and D. This variation trend is reversed in sample A or B. The versatile variation trends of the spectral peak of radiated power can be attributed to the interplay between two counteracting factors. First, when the distance between a radiating dipole and the major portion of electron oscillation for SP coupling on a metal NP increases, the spectral peak of coupled radiated power is red-shifted due to the change of interference condition between the radiations of the LSP resonance and the dipole itself [5, 16, 17]. The second factor is the blue-shift trend due to the upward extension of surface charge or mode field distribution of the coupled LSP mode into the NP portion surrounded by air (lower refractive-index) when the *x*-coordinate of an *x*-oriented dipole increases, as shown in Figs. 14(a)-14(d). The interplay between the red- and blue-shift factors leads to various shifting results in different samples.

The dielectric structure of sample D is designed for comparing the SP coupling behavior of a metal NP with dielectric coating to those of the other two dielectric structures. For this purpose and for simplifying numerical computation, we assume that the dielectric-coated Ag NP has the same shape as those in the other two samples. However, the real geometry of a dielectric-coated metal NP can be closer to a sphere. Compared to the geometry in Fig. 1(d), in which the coupled LSP mode field distributes mainly around the NP base, the major LSP mode field distribution in a spherical dielectric-coated metal NP on GaN is farther away from the radiating dipole in a QW. In this situation, the SP coupling effect in a structure with a spherical dielectric-coated metal NP is expected to be weaker. Also, the spectral peak of SP-coupled radiated power can be further blue-shifted in a structure with a spherical dielectric-coated metal NP because the major LSP mode field distribution is further away from the region of high refractive-index, i.e., GaN.

## 5. Conclusions

In summary, we have numerically compared the SP coupling behaviors of an embedded radiating dipole in GaN with a surface Ag NP in four structures of different added dielectric geometries, including an extended DI and a DI of a finite width between the Ag NP and GaN, a dielectric coating on the Ag NP, and no dielectric addition. Either an added DI or dielectric coating could lead to the blue shift of LSP dipole resonance peak or the spectral peak of radiated power enhancement ratio with respect to that of the structure without dielectric addition. A smaller dielectric refractive-index or a larger dielectric thickness resulted in a larger blue-shift range. Under the condition of the same dielectric refractive-index and thickness, the structure of a DI with a finite width led to the largest blue-shift range, followed by the structure of an extended DI and then the structure of dielectric coating. Our study also showed that although the LSP resonance peak could be blue-shifted by reducing the size of a surface Ag NP, its SP coupling strength was dramatically reduced. Adding a DI or dielectric coating is a more practical approach for shifting the major LSP resonance mode of a surface Ag NP from the green into blue range.

## Acknowledgments

This research was supported by Ministry of Science and Technology, Taiwan under the grants of MOST 103-2221-E-236-006, MOST 103-2221-E-002-139, NSC 102-2221-E-002-204-MY3, and MOST 103-2120-M-002-002, and by the Excellent Research Projects of National Taiwan University (103R890951 and 103R890952).

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